Abstract
The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] to make [Formula: see text] strong rainbow vertex connected. These two concepts are inspired by the concepts of proper (strong) rainbow connection number of graphs. In this paper, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text] and [Formula: see text]. Moreover, we determine the values of [Formula: see text] and [Formula: see text] of [Formula: see text] with large clique numbers [Formula: see text], [Formula: see text] and [Formula: see text].
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